A simple presentation of the effective topos

Alexis Bernadet (1) , Stéphane Graham-Lengrand (1, 2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 PARSIFAL - Proof search and reasoning with logic specifications

We propose for the Effective Topos an alternative construction: a realisability framework composed of two levels of abstraction. This construction simplifies the proof that the Effective Topos is a topos (equipped with natural numbers), which is the main issue that this paper addresses. In this our work can be compared to Frey's monadic tripos-to-topos construction. However, no topos theory or even category theory is here required for the construction of the framework itself, which provides a semantics for higher-order type theories, supporting extensional equalities and the axiom of unique choice.

Mots clés: category; topos; realisability; type theory; program extraction

Domaines: Logique en informatique [cs.LO]

Alexis Bernadet, Stéphane Graham-Lengrand. A simple presentation of the effective topos. [Research Report] LIX, Ecole polytechnique. 2012. ⟨hal-00844250⟩